Cosmological Perturbations and Degrees of Freedom

A homogeneous solution can hide precisely the modes that decide whether an alternative theory is viable. Cosmological perturbation theory reveals those modes by separating coordinate freedom and constraints from propagating scalar, vector, and tensor variables. The decisive object is the reduced quadratic action: its kinetic and gradient matrices determine ghosts, characteristic speeds, and the domain in which linear predictions are meaningful.

Helpful background. Einstein Equations supplies the linearized constraint/evolution split; Gravitational Actions and Boundary Terms supplies the second-variation method; Formulations of General Relativity supplies gauge and constraint counting; Gravity Beyond General Relativity supplies the extra-field and effective-theory scope; Causality and Global Structure supplies the characteristic-cone interpretation; and Gravitational Radiation and Waveforms supplies the transverse-traceless tensor normalization.

Scalar, vector, and tensor variables

Write the spatially flat background in conformal time, $ds^2=a^2(-d\eta^2+d\boldsymbol{x}^2)$. A general linear metric perturbation decomposes as

$$\delta g_{00}=-2a^2 A, \qquad \delta g_{0i}=a^2(\partial_i B+S_i),$$ $$\delta g_{ij}=a^2\left[ 2H_L\delta_{ij} +2\left(\partial_i\partial_j-\frac13\delta_{ij}\nabla^2\right)E +2\partial_{(i}F_{j)}+h_{ij} \right].$$

Here $\partial_iS_i=\partial_iF_i=0$, while $\partial_i h_{ij}=0$ and $h_{ii}=0$. Rotational symmetry makes the scalar, vector, and tensor sectors independent at linear order. A scalar coordinate transformation reshuffles $A,B,H_L,E$; it does not remove a physical scalar supplied by the theory. Gauge-invariant Bardeen potentials may be defined so that in Newtonian gauge, $B=E=0$, the scalar line element is

$$ds^2=a^2\left[-(1+2\Psi)d\eta^2+(1-2\Phi)d\boldsymbol{x}^2\right].$$

This representation defines the signs of $\Psi$ and $\Phi$ used in the following pages. It should be reached only after writing the covariant equations or gauge-invariant combinations; setting variables to zero prematurely can erase a constraint equation.

Reduce the constraints before counting modes

Expand the complete gravity-plus-matter action to second order about a background that satisfies its own equations. Variables without time derivatives—usually lapse and shift perturbations—are auxiliary. Vary them, solve their linear constraint equations when the constraint matrix is invertible, and substitute the solutions back. For $N$ remaining scalar amplitudes collected in $\boldsymbol q$, a Fourier-mode action can be organized as

$$S^{(2)}_{\mathrm s}=\frac12\int dt\,d^3k\,a^3 \left[ \dot{\boldsymbol q}^{\mathsf T}K\dot{\boldsymbol q} -\frac{k^2}{a^2}\boldsymbol q^{\mathsf T}G\boldsymbol q -\boldsymbol q^{\mathsf T}M\boldsymbol q +2\dot{\boldsymbol q}^{\mathsf T}B\boldsymbol q \right].$$

The symmetric kinetic matrix $K$ must be positive definite on the regime being claimed. A negative eigenvalue is a ghost: the Hamiltonian is unbounded within the effective description. At wavelengths short compared with background and mass scales, the squared propagation speeds solve

$$\det\!\left(c_s^2 K-G\right)=0.$$

A negative root is a gradient instability whose rate grows with physical wavenumber until the effective cutoff intervenes. This differs from a negative effective mass squared in a canonically normalized, approximately stationary mode: its tachyonic growth is limited to long wavelengths and can be slow compared with the background evolution. In a coupled time-dependent system, a negative eigenvalue of the displayed $M$ alone is not a sufficient diagnosis. Canonical normalization, expansion, and the $B$ term all affect the finite-frequency equation.

The order of operations matters. Before auxiliary variables are eliminated, an apparent zero or negative kinetic entry can belong to a constraint rather than a propagating mode. Conversely, solving a constraint by dividing by a coefficient that vanishes on a special branch can hide strong coupling. The quadratic-action method for general single-scalar dark-energy theories is organized in Bellini and Sawicki 2014, §§ 3.1 and 3.3, Eqs. (3.12)–(3.14), and Appendix A.

For an elementary example, take $K=\operatorname{diag}(2,1)$ and $G=\operatorname{diag}(2,-1/4)$. Both modes have positive kinetic energy, but their squared speeds are $1$ and $-1/4$. The second mode grows as $\exp[kt/(2a)]$ when coefficients are frozen on a short time interval. A correct mode count and positive kinetic terms therefore do not establish a usable evolution problem; the unstable short-wavelength sector also lacks real wave characteristics in this two-derivative description.

Sector-by-sector interpretation

In general relativity with ordinary perfect-fluid matter, the metric vector sector carries no independent vacuum wave and decays in an expanding universe. Preferred-frame vector theories can instead supply transverse vector polarizations; their vector kinetic and gradient coefficients require separate checks. The tensor sector contains the two helicities of $h_{ij}$ in general relativity, while massive or multimetric models can mix them with additional tensor amplitudes.

For a single tensor field with no parity violation, the reduced action has the schematic form

$$S^{(2)}_{\mathrm T}=\frac18\int dt\,d^3x\,a^3 M_T^2 \left[\dot h_{ij}\dot h_{ij} -\frac{c_T^2}{a^2}\partial_kh_{ij}\partial_kh_{ij} -m_T^2 h_{ij}h_{ij}\right].$$

Thus $M_T^2\gt0$ is the tensor no-ghost condition and $c_T^2\gt0$ is the short-wavelength gradient condition. The mass, friction arising from $a^3M_T^2$, and possible mixing are logically distinct. Their observable propagation effects are derived on the gravitational-wave page.

The scalar sector is usually the most model-dependent. A perfect fluid contributes its density mode; a scalar–tensor theory adds a gravitational scalar; higher-derivative degeneracy may remove a would-be Ostrogradsky mode. Counting fields in the Lagrangian is therefore insufficient. One must count the rank of the reduced kinetic matrix on the selected background.

Limits of the quadratic test

Stability is scale and time dependent. A matrix that is positive today may cross a zero eigenvalue in the past, and a high-$k$ conclusion is relevant only below the theory’s physical cutoff. At $k=0$, the scalar–vector–tensor decomposition and constraint rank can differ from their finite-$k$ limits; homogeneous perturbations must be treated separately. A positive quadratic action also does not prove nonlinear stability or a well-posed nonlinear initial-value problem.

These qualifications are central to cosmological parameterizations. The linear theory must state its background branch, wavenumber interval, time interval, matter model, and cutoff. Koyama’s review shows how these stability and scale conditions enter concrete modified-gravity cosmologies in Koyama 2016, §§ 2.3, 3.2, and 4.

Exercises

Suppose two reduced scalar modes have

$$K=\begin{pmatrix}A&C\\ C&B\end{pmatrix}, \qquad A\gt0.$$

Find the remaining no-ghost condition without explicitly diagonalizing $K$, and explain what happens at equality.

Solution

Sylvester’s criterion says that a real symmetric $2\times2$ matrix is positive definite exactly when its leading principal minors are positive. Since $A\gt0$ is given, the second condition is

$$\det K=AB-C^2\gt0.$$

Equivalently, after completing the square,

$$A\left(\dot q_1+\frac{C}{A}\dot q_2\right)^2 +\left(B-\frac{C^2}{A}\right)\dot q_2^2$$

has two positive coefficients. At $AB=C^2$, one kinetic eigenvalue vanishes. That may signal an additional constraint on a special branch, but generically it marks strong coupling: interactions omitted from the quadratic action are no longer suppressed by a nonzero canonical kinetic term.

References

• Bellini, Emilio, and Ignacy Sawicki. “Maximal Freedom at Minimum Cost: Linear Large-Scale Structure in General Modifications of Gravity.” Journal of Cosmology and Astroparticle Physics 2014 (07), 050 (2014). DOI. Open PDF
• Koyama, Kazuya. “Cosmological Tests of Modified Gravity.” Reports on Progress in Physics 79, 046902 (2016). DOI. Open PDF